# Questions tagged [uniform-spaces]

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### Delone sets in terminology of uniformities

I've been reading about Delone sets recently, and noticed that they are formulated either in terms of topological groups or in terms of metric spaces. Group terminology: Let $G$ be a locally compact ...
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### Do compact inverse-property loops (or just compact Moufang loops) have bi-invariant Haar measure?

So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left invariant? (And we can restrict to Moufang loops if ...
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### Topology generated by complete and incomplete uniformities [closed]

Does there exist a topology which can be induced simultaneously by a complete and an incomplete uniformity?
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### Convergent net in a quasi-uniform space which is not Cauchy

The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a ...
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### Results that are easier in a metric space

Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces? In particular, I'm ...
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Let $X$ be a uniform space and $F(X)$ the vector space of all uniformly continuous real-valued functions over $X$. It is possible to express every bounded uniform semimetric $d$ on $X$ as $d(x,y) = ... 1 vote 0 answers 89 views ### In a topological group$G$with its lower uniformity, if$G$is locally totally bounded, is its completion locally compact? There has been work done on groups whose lower uniformity (or Roelcke uniformity) is totally bounded, e.g. the orthogonal group on a Hilbert space. This condition is equivalent to saying the lower ... 5 votes 0 answers 130 views ### Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded? Is there an example of an Abelian group$G$and group topologies$\cal S$and$\cal T$on it such that$\cal S$is an immediate successor to the trivial topology on$G$(i.e there is no other group ... 11 votes 2 answers 1k views ### Baire Category Theorem for complete uniform spaces The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ... 4 votes 1 answer 254 views ### In the category of uniform spaces, is the completion of a quotient map also a quotient map? I asked this question about 2 months ago on math.stackexchange, but so far I received neither comments nor answers. Let$X$and$Y$be two Hausdorff uniform spaces. A surjective uniformly continuous ... 4 votes 1 answer 292 views ### The subbase theorem for total boundedness The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) : Let$(X,\mathcal{U})$be a uniform ... 3 votes 1 answer 140 views ### Totally bounded group topologies on$\Bbb Q$with trivial intersection of two neighborhoods Are there totally bounded group topologies$\mathcal S$and$\mathcal T$on$\Bbb Q$such that for some open sets$A\in\mathcal S$and$B\in \mathcal T$we have$A\cap B=\{0\}$? 11 votes 1 answer 247 views ### Duality between large and small scale structures A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure$\mathcal{C}$(defined by a ... 4 votes 0 answers 261 views ### The Haar integral on uniform spaces Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability. As ... 4 votes 1 answer 2k views ### How is the notion of a Lipschitz structure on a manifold defined? According to wikipedia, there is such a definition.$\:$The candidate that I can come up with is "an equivalence class of metrics that induce the topology and make the space locally bi-Lipschitz to ... 5 votes 2 answers 198 views ### A theorem of Markov about completely regular spaces and topological groups In Pontriaguin's classic book Grupos continuos (in English Continuous Groups), says that A. Markov proved that: There are topological groups that are not normal. Furthermore, he says it is deduced ... 1 vote 1 answer 98 views ### Existence of a moderate uniform structure on$\Bbb R$A moderate uniform structure$\mathcal U$on$\Bbb R$is one for which$\forall U\in \mathcal U, \exists n\in \Bbb N,\quad U^n=\Bbb R^2$but$ \not\exists n\in \Bbb N,\forall U\in \mathcal U,\quad U^... 200 views

### Is there a normal space that is not uniformly normal

Let $(X,\mathcal D)$ be a uniform space and $A,B\subseteq X$. Let's say $A$ is uniformly inside $B$ and write $A\le B$ iff there's some entourage $D$ for which $$(\forall a\in A)(D[a]\subseteq B)$$ A ... 1 vote
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Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase $$\Lambda =\{ \{(f,... 2 votes 1 answer 368 views ### Extend Homeomorphism to Uniformly Continuous Function I have a space A which is homeomorphic to the open n-ball B_n. I'm trying to build a CW-complex with it, so I want a continuous function from the closed ball \overline{B}_n to the closure \... 1 vote 1 answer 158 views ### Precompact reflection in diagonal uniform spaces Each diagonal uniform space (X,\mathcal D) can be derived from the covering uniform space (X,\Sigma_{\mathcal D}) and each covering uniform space (X,\Sigma) can be derived from the diagonal ... 1 vote 1 answer 119 views ### Reference: uniformity of pointwise convergence has no countable base Does anyone have a reference for the fact that the uniformity of pointwise convergence on real functions of [0,1] (that is, the uniformity generated by the sets \lbrace (f,g) : |f(x) - g(x)| < \... 3 votes 3 answers 389 views ### Complete uniform spaces require complete metrics? Hey all, It is well-known that any uniformity is generated by the family of pseudometrics which are uniformly continuous from the product uniformity to \mathbb{R}. Further, the uniformity is ... 2 votes 2 answers 269 views ### Uniformities generated by metrics. Any uniformity on a set X is generated by a family of pseudometrics on X. So if (X,\mathcal D) is a uniform space there's a set P of pseudometrics on X with$$\mathcal D=\left< \bigcup_{... 1 vote
This is a follow up question to this one. If $X$ is a metric space, denote by $C_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-norm). Do we have $... 2 votes 1 answer 1k views ### Extending uniformly continuous functions on subspaces to non-metrizable compactifications I have a complete metric space$Y$, some non-metrizable(!) Hausdorff compactification$Z$of it and a subspace$X \subset Y$. Furthermore, I do have a uniformly continuous function$f$on$X$. So ... 3 votes 2 answers 216 views ### For any entourage$U,V$there's an entourage$W$such that$U\circ W\subseteq V\circ U$Let$(X,\mathcal U)$be a uniform space and let$U\in \mathcal U$. Is this statement true? $$\forall V\in \mathcal U, \exists W\in \mathcal U, U\circ W\subseteq V\circ U$$ I think if the above ... 2 votes 1 answer 550 views ### A uniformity with a countable base is a pseudometric uniformity. I need a proof for this proposition: If a uniformity$\mathfrak U$on$X$has a countable fundamental system of entourages, then it can be defined by a pseudometric on$X$. which is the ... 2 votes 1 answer 234 views ### Does every proximal outer measure, measure all open sets? Let$\: \langle X,\delta\rangle \: $be a separated proximity space. Let$\: \mu^* \: : \: 2^{X} \: \to \: [0,+\infty] \: $be a proximal outer measure. Let$U$be an open subset of$X$. Does it ... 12 votes 2 answers 607 views ### Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE] And what else can be said, if so? (Original math.SE post) In more detail: Say$(G,\mathscr{T})$is a topological group. It has a left uniformity$\mathscr{L}$and a right uniformity$\mathscr{R}$. (... 4 votes 3 answers 467 views ### Does every Lindelof uniform space have a Lindelof completion? Recall Lindelof = every open cover has a countable subcover. Is the question of the title answered by known material from some standard text on uniform spaces? Note it is well known to be true for ... 4 votes 1 answer 279 views ### Does the weak approximation theorem hold for general topological fields? The weak approximation theorem states that given a field$F$and nontrivial inequivalent absolute values$|\cdot|_1,\ldots,|\cdot|_n,$and letting$F_i$denote$F$with the topology from$|\cdot|_i$, ... 7 votes 4 answers 2k views ### Finite dimensional vector spaces over a complete but not-necessarily-valued field I'm essentially reopening this old question of Ricky Demer which was never fully answered. Essentially the original question: Suppose we have a topological field$F$which is complete, Hausdorff, and ... 3 votes 1 answer 365 views ### Chaos in uniform spaces Let$Dom$be a uniform space, and$\hspace{.04 in}f$be a continuous function from$Dom$to itself satisfying: For all non-empty open subsets$U$and$V$of$Dom$, there exists a natural number$n\$ ... 